As shown in appendix ,
the graph polynomials are now
[box=]align

U = |g|,       F= tr(adj(g)G),       FU = tr(g-1G),

where \(|g|=\mathrm {det}\,g\) , \(\mathrm {adj}\,g=|g| \,g^{-1}\)  is the adjugate matrix and \(g^{-1}\)  the inverse matrix.
The derivation proceeds by expressing the delta functions of () in Fourier form and integrating out all internal momenta. Only after this step has been performed are the Fourier integrals for the delta functions then evaluated.
As the Gram determinant \(|G|\)  is proportional to the squared volume of the simplex spanned by the independent momenta, the polynomial \(\mathcal {U}\)  describes the image of this squared volume under the mapping \(V_{ij}\rightarrow v_{ij}\) . Alternatively, by the matrix tree theorem (see e.g., [1]}), \(\mathcal {U}\)  is the Kirchhoff polynomial encoding the sum of spanning trees on the simplex.
